2.1 System setup

The system structure of filled-aperture CBC based on deep learning phase control is depicted in Figure 1. The seed laser is split into multiple beam channels by a beam splitter (BS), and each beam path passes through a phase modulator (PM), amplifier and collimator (CO). All beams are tiled in a dense array, such as square shape^[ Reference Bourderionnet, Bellanger, Primot and Brignon ^{25 ]}, hexagon shape^[ Reference Ma, Chang, Ma, Su, Qi, Wu, Li, Long, Lai, Chang, Hou, Zhou and Zhou ^{15 ]} or others^[ Reference Li, Liao, Gao, Sun, Tan, Wang and Lan ^{41 ]}, and are then emitted in the same direction. The laser array is sampled before coming into the filled-aperture beam combiner to obtain the tiled-aperture combined pattern by a camera, which is input into the neural network to make phase prediction and compensate phase noise with high-bandwidth operation. The combiner can take a variety of forms, such as diffractive optical element (DOE)^[ Reference Thielen, Ho, Burchman, Goodno, Rothenberg, Wickham, Flores, Lu, Pulford, Robin, Sanchez, Hult and Rowland ^{42 ]}, intensity BS-based binary tree or segmented mirror^[ Reference Müller, Aleshire, Klenke, Haddad, Légaré, Tünnermann and Limpert ^{16 ]}, which transforms spatially separated beams into an overlapped beam. The segmented mirror combiner is employed in this paper since it is easy to combine a laser array of linear or square tiling, which will be superimposed into a single beam at one output port if amplitudes match the splitting ratios and phases are locked; it is only represented by a box in Figure 1 for simplicity, and its specific architecture can be found in Ref. [Reference Aleshire, Steinkopff, Jauregui, Klenke, Tünnermann and Limpert43]. A liquid crystal (LC) is inserted in the sampled path of each spatially separated beam, so the optical path delays of all beams can be artificially adjusted. The laser array of main power illuminates on the beam combiner and is combined into a single aperture beam in the near field, and the central intensity of the combined beam is sampled by a photodetector (PD), which is used to drive the delay controller and monitor the combining efficiency.

Figure 1

System setup of deep learning phase control for filled-aperture CBC.

As the beam combiner needs to fill the aperture, each beam passes through its own combining route, which results in different optical paths and laser phases. Therefore, the phase relationships at the camera are not consistent with those at the PD due to optical path differences, which accounts for why the delay control is essential to compensate for the static path-related phases. It should be noted that although the optical path difference dominantly occurs inside the combiner, it can be eliminated by adjusting the phase delays in the sampled path instead, which is consistent with high-power operation due to the avoidance of manipulation on high-power beams. The delay control is an optimization problem of multiple variables, and thus stochastic parallel gradient descent (SPGD) can be employed to address the issue conveniently. The process of delay control is described in the flow chart in Table 1, where T _p/T _d denotes the time step of phase/delay update, and γ and σ are the gain coefficient and perturbation amplitude, respectively, of the SPGD algorithm. As the SPGD algorithm requires two-way perturbations before parameter update, one step of delay control is divided into three mini-steps, so the current step of delay control is calculated according to T _d/3, as shown in line 4 of Table 1. The phase control is much faster than delay control (T _p ≪ T _d), so phase-locking is achieved immediately after the delays are perturbed or updated. It should be emphasized that the PD signal does not change directly with delays of LCs but changes with phases of PMs, since the neural network will dynamically drive PMs to compensate for the delay changes and maintain phase-locking at the camera.

Table 1

Procedure for delay control.

With the delay control done, the phase relationships at the tiled-aperture path (camera) are the same as at the filled-aperture path (PD), and thus the filled-aperture combining efficiency will be maximized when laser phases are synchronized at the camera by the neural network. Therefore, filled-aperture phase control can be converted into tiled-aperture phase control with proper delay control. In addition, optical path length differences in the combiner are generally piston errors, so the delay control can be implemented in a once and forever manner theoretically, and only phase control is required finally.

2.2 Training strategies

Although the mapping from the combined pattern to laser phases can be interpreted as a pattern recognition problem, there are two barriers to cross when applying deep learning to the phase control of CBC. The first is the phase periodicity-induced discontinuity, where φ and φ + 2kπ (k = 0, ±1, ±2, ⋯) represent the same phase, but their distance is 2kπ. In general, phase is limited to a single period [–π, π) so that every value will correspond to a unique phase, but this does not work well in the training of neural networks as the distance between label and output is usually used to calculate loss. For instance, if the network predicts 0.9π as –π, the prediction is actually good because it is equivalent to –1.1π and –π with a real distance of 0.1π, but it will result in a large loss because the network interprets the distance as 1.9π. To deal with this, a sin-cos loss is introduced and defined as follows:

(1) $$\begin{align}L=\frac{1}{2M\left(N-1\right)}\sum \limits_{m=1}^M\sum \limits_{n=1}^{N-1}&\left[{\left|\sin {\varphi}_n^{(m)}-\sin {\varphi}_n^{\ast (m)}\right|}^2\right.\nonumber\\& \ \ \left.+{\left|\cos {\varphi}_n^{(m)}-\cos {\varphi}_n^{\ast (m)}\right|}^2\right]\end{align}$$

where φ* and φ are the true phase and the predicted phase, respectively, subscript n denotes the beam number (the total is N–1 as one beam is set as reference) and superscript m denotes the sample number (the total is batch size M).

The proposed loss function calculates both sine and cosine distances between the network output and true phase, which is inducive to overcome the aforementioned phase discontinuity, since phase is not compared directly. However, if the network is requested to output phase predictions φ, the discontinuity remains unsolved. More specifically, phases of around –π and π should be predicted using similar image features with the same network parameters, so the network will struggle to minimize losses at both ends rather than learning an accurate and unique mapping function, which means that the learning problem is not a strict one-to-one mapping, so the learning process is not very good. Therefore, the output of the network is designed to consist of both the sine and cosine of phase φ. It should be noted that the sine and cosine values may not strictly satisfy the equation of sin² φ + cos² φ = 1, as they are predicted independently by the neural network. Letting C and S represent the predicted cosine of phase (cosφ) and predicated sine of phase (sinφ), respectively, although C and S probably correspond to different predicated phases, a more accurate phase prediction φ can be calculated using both C and S by Equation (2). By doing so, the prediction discontinuity can be solved, as the real part and imaginary part are both continuously varied, and phases of –π and π are positioned in the same line of the complex plane:

(2) $$\begin{align}\varphi =\mathrm{angle}\left(C+ iS\right).\end{align}$$

The other barrier in the application of deep learning phase control is the non-unique phase-intensity mapping, which represents that different phase relationships may result in the same intensity distribution of the far field. Many solutions have been developed to solve the non-unique mapping problem, and one of the simplest, a diffractive pattern at a slightly defocused plane, is adopted here. In the simulation, all beams are assumed to be Gaussian fundamental mode and linearly polarized in the same direction. The intensity of the combined beam at the defocus far field can be expressed as follows:

(3) $$\begin{align}I\left(\boldsymbol{r},z=L\right)=\ &\frac{1}{i\lambda L}\exp \left(i\frac{\pi {r}^2}{\lambda L}\right) \mathrm{FT}\nonumber\\& \times \left\{\vphantom{\left[-{\left(\frac{\pmb{\rho} -{\pmb{\rho}}_{0n}}{w}\right)}^2\right]}\exp \left[i\frac{{\pi \rho}^2}{\lambda}\left(\frac{1}{L}-\frac{1}{f}\right)\right]\right.\nonumber\\& \left.\times\sum \limits_{n=0}^{N-1}A\exp \left[-{\left(\frac{\pmb{\rho} -{\pmb{\rho}}_{0n}}{w}\right)}^2\right]\exp \left(i{\varphi}_n\right)\right\},\nonumber\\[-10pt]\end{align}$$

where A, φ, w, $\pmb{\rho}_0 $ and λ are the amplitude, phase, waist width, central position and wavelength of the laser, respectively, and $\pmb{\rho} $ and $\boldsymbol{r} $ are the position vectors at the emission plane and observation plane while ρ and r of normal forms represent corresponding scalar distances from the origin. FT{·} refers to Fourier transform, and L and f represent the propagation length and the focus length, respectively (in our study, L ≠ f). It should be pointed out that the strategy of the defocus pattern is not contributed in this work, but it is important not only for solving the non-unique mapping problem but also for clarifying the optical model of dataset preparing. Therefore, the advantages of the strategies discussed in the following section correspond to the strategies of sin-cos loss and two-layer output, rather than the defocus pattern, since the training dataset is kept the same with or without strategies.

2.3 Neural network construction

The VGG model, which is a powerful and prevalent tool in computer vision, is modified and employed as our neural network to achieve CBC phase control^[ Reference Simonyan and Zisserman ^{44 ]}. It is a classical convolution neural network, which consists of a series of convolution layers, max-pooling layers and fully connected (FC) layers, as shown in Figure 2. For simplicity, the network is divided into five blocks, and each block is made up of one or more convolution layers followed by a max-pooling layer. Based on the number of convolutional layers, VGG can be named VGG-11, VGG-13 and so on, and the one used in our study is VGG-16, including 13 convolution layers and three FC layers. The kernel size of all convolution layers is 3×3 and the padding size is 1, so the image height and width do not change after convolution, but the channel number might be different depending on the number of convolution filters. The kernel size of max-pooling layers is 2×2, so the image height and width will be reduced to the half after pooling. It should be noted that a rectified linear unit (ReLU) function is followed with each convolution layer and FC layer, which are not shown in the figure. A dropout layer with a possibility of 0.5 is added in both FC1 and FC2 to reduce the possibility of overfitting, and the activation function of FC3 is changed from softmax to sigmoid, as phase prediction is a regression problem rather than a classification problem.

Figure 2

Structure chart of the VGG network.

The built VGG network can extract features of the input intensity profile by convolution operations layer by layer, and it can leave out redundant features by max-pooling layers to prevent overfit. In addition, the ReLU activation layers are beneficial to enhance the nonlinear expression ability of the network. After a series of convolution and max-pooling layers, the image features are efficiently extracted and condensed, and are flattened and reorganized by the FC layers. Finally, the output layer is formatted in size of 2 × (N – 1) while it is 1 × (N – 1) for traditional networks, so it maps these extracted features into the real and imaginary parts of a complex phase vector and achieves phase prediction consequently. To accommodate the phase-control tasks, the input image size of the VGG-11 model is changed to 64×64×1, and the intensity value is normalized with the maximal value being 1.